17 questions linked to/from The subring test
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### How to prove a set is an integral domain?

Let's say I have the ring: $\mathbb{Z}[\sqrt{2}]=\{a+b\sqrt{2}\mid a,b\in\mathbb{Z}\}$. Now the question asked is to prove whether or not this ring is an integral domain. By definition: "An ...
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### Center is always a commutative subring [duplicate]

Let R be a ring. The center of R is defined to be the set of all elements x ∈ R such that xr=rx for all r ∈ R. Prove that the center of R is always a commutative subring of R. I understand how to ...
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### Prove that the center of a ring is a subring.

The center of a ring $R$ is $\{c\in R : cr=rc$ for every $r \in R\}$. Prove that the center of a ring is a subring. What is the center of a commutative ring? Is my solution right? solution You ...
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### Subring Test for non-empty subset $S ⊂ R$ [closed]

Show that a non-empty subset $S ⊂ R$ is a subring of $R$ if for all $r, s ∈ S$ we have $r − s ∈ S$ and $rs ∈ S$. (This makes it easier to verify a set is a ring, if you know the set lives in a larger ...
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### Proof Verification: Let S be a non-empty subset of a ring R. Then S is a subring of R if and only if S is closed under $-$ and $\times$.

Let S be a non-empty subset of a ring R. Then S is a subring of R if and only if S is closed under $-$ and $\times$. Proof: First, prove that S is a subgroup of R. Pick an arbitrary element $x$ from ...
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### Showing a subset of the functions $\mathbb R \to \mathbb R$ is a subring

Let $T$ be the set of all functions from $\mathbb R$ to $\mathbb R$. Let $S = \{ f \in T : f(2) = 0 \}$. Show that $S$ is a subring of $T$. $T$ has a zero element and an identity element by ...
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### Prove the following is a subring

Let $R=\{m+n\sqrt{2} \mid m,n\in \mathbb{Z}\}$. Prove that $R$ is a subring of the real numbers. I just want to know how to get started really. My professor has used the same example for the past ...
### Prove that $(n\mathbb Z, +, \times )$ are the only subrings of $(\mathbb Z, +, \times)$
I had to find all the subrings of the integers and then prove that there aren't any more. It's clear to me the $(n\mathbb Z, +, \times )$ is a subring of the integers for all $n$ element of the ...